Download Seminars in Cancer Biology Invasion emerges from cancer cell

Seminars in Cancer Biology 18 (2008) 338–348
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Seminars in Cancer Biology
journal homepage: www.elsevier.com/locate/semcancer
Review
Invasion emerges from cancer cell adaptation to competitive microenvironments:
Quantitative predictions from multiscale mathematical models
Vito Quaranta a,∗ , Katarzyna A. Rejniak b , Philip Gerlee b , Alexander R.A. Anderson b,∗∗
a
b
Department of Cancer Biology, Vanderbilt University School of Medicine, Nashville, TN 37235, USA
Division of Mathematics, University of Dundee, Dundee DD14HN, Scotland, UK
a r t i c l e
i n f o
Keywords:
Cancer
Invasion
Mathematical modeling
Adaptation
Microenvironment
Evolution
Cancer progression
Selection
Computer simulations
Cancer phenotypes
Darwinian selection
a b s t r a c t
In this review we summarize our recent efforts using mathematical modeling and computation to simulate
cancer invasion, with a special emphasis on the tumor microenvironment. We consider cancer progression
as a complex multiscale process and approach it with three single-cell-based mathematical models that
examine the interactions between tumor microenvironment and cancer cells at several scales. The models
exploit distinct mathematical and computational techniques, yet they share core elements and can be
compared and/or related to each other. The overall aim of using mathematical models is to uncover the
fundamental mechanisms that lend cancer progression its direction towards invasion and metastasis.
The models effectively simulate various modes of cancer cell adaptation to the microenvironment in
a growing tumor. All three point to a general mechanism underlying cancer invasion: competition for
adaptation between distinct cancer cell phenotypes, driven by a tumor microenvironment with scarce
resources. These theoretical predictions pose an intriguing experimental challenge: test the hypothesis
that invasion is an emergent property of cancer cell populations adapting to selective microenvironment
pressure, rather than culmination of cancer progression producing cells with the “invasive phenotype”.
In broader terms, we propose that fundamental insights into cancer can be achieved by experimentation
interacting with theoretical frameworks provided by computational and mathematical modeling.
© 2008 Elsevier Ltd. All rights reserved.
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical frameworks of cancer based on multiscale mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Approaches and techniques to modeling cancer invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The mE in three mathematical models: EHCA, IBCell and HDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Brief description of the EHCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Brief description of IBCell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Brief description of HDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distinctive mE modeling features of EHCA, IBCell and HDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Insights from simulated tumor growth dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.
Fingering morphology as a modeling abstraction of invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.
mE influence on tumor evolutionary dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.
Adaptation and selection during cancer progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.
Parallels between cancer progression and Darwinian evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗ Corresponding author. Tel.: +1 615 936 2868.
∗∗ Corresponding author. Tel.: +44 1382 384462.
E-mail addresses: [email protected] (V. Quaranta),
[email protected] (A.R.A. Anderson).
1044-579X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.semcancer.2008.03.018
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1. Introduction
We are studying the process of cancer progression, particularly
invasion, by a multidisciplinary approach that integrates mathematics and experimentation. In this review we describe why cancer
progression and invasion are amenable to this approach, the logic
behind the modeling techniques we are using in our group, and
key outcomes our mathematical models have in common. The
emphasis of the review is on the tumor microenvironment (mE),
for two reasons. First, this is the theme of this journal issue. Second,
three independent computational models we developed, based on
distinct mathematical techniques, all point to an essential, but
unsuspected role for the mE in eliciting invasive patterns of tumor
growth and enabling dominance of aggressive cell phenotypes.
2. Theoretical frameworks of cancer based on multiscale
mathematical models
Cancer progression can be viewed as a process with both multiscale and complex features. Multiscale because its mechanisms
and outcomes involve multiple biological scales, e.g., from genes
to cells to tissues to organs to organisms to populations. Complex because it is influenced by multiple variables interacting at
once and segregated by scales; e.g., at the population scale, variables may include age-specific rate of cancer incidence, exposure
to chemicals, lifestyle; at the tissue scale, variables may include
rates of cell proliferation, death, motility, adhesion, angiogenesis;
at the molecular scale, mutation rates.
In a complex multiscale process, variables at a given scale affect
properties of the process at other scales. They tend to do so not
in a linear fashion but, rather, from underlying new properties
that emerge at the next higher scale. Applying this perspective to
cancer progression, invasion can be viewed as a property of the
tissue scale that emerges from the population behavior of individual cells at the lower scale. Of course, individual cell behavior
is influenced by the mE, a composite of molecular, cellular and
tissue scales that illustrates how biological scales intermingle. Ideally, in a comprehensive mechanistic view of cancer progression,
all scales should ultimately interact with one another, in both
directions, with the cell as the centerpiece for these interactions
(see below). Whilst this interactivity is easily inferred from current knowledge on cancer progression, at the moment studies and
insights at different scales remain quantitatively disconnected. For
example, major insights from epidemiological data and the study of
age-specific cancer incidence (i.e., the population scale) led to the
formulation of a multistage theory [1,2] predicting that there are
five to seven rate limiting steps in cancer progression. Validating
this theory and understanding the nature of these steps at lower
scales is a major goal of current research that has led to spectacular advances, but that remains largely unrealized. Another success
story is the case of familiar retinoblastoma two-hit theory [3] (a
special case of multistage progression), which predicted the existence of an Rb gene, subsequently discovered by molecular genetics
[4]. It remains however to be determined how altered functions
of mutated Rb genes propagate through higher scales (cell, tissue,
etc.) to eventually cause the observed retinoblastoma incidence at
the population level. Discovery of other tumor suppressor genes
was promoted by multiscale links from population to molecular
genetics studies [5], but again the intermediate cell scales remain
largely unexplored. Conversely, in non-familiar sporadic cancers,
fundamental insights have been obtained starting from the gene
scale [5]. Molecular cancer genetics has identified oncogenes that,
together with tumor suppressor genes, can reproduce many aspects
of cancer progression in cultured cells or animals. In these experi-
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mental models, however, cancer progression is initiated by genetic
mutation, but again steps at intermediate scales (e.g., at the cellular and tissue scale), other than unrestrained growth, are not well
reproduced or understood quantitatively. For instance, invasion and
metastasis rarely occur in animal models of human cancers, suggesting that other variables or mechanisms may be involved. At the
cell scale, the realization that changes in signaling networks may
confer cancerous traits to a cell has produced major insights. However, defining the link between these traits and underlying genetic
changes is a major challenge [6] to be met. At the tissue scale the
chemical, physical and cellular composition of the mE is known to
profoundly affect cancer progression [7]. In some cases, mE features
appear to dominate over genotype in determining the outcome of
cancer cell organization into tissue [8].
A delaying factor is that, as studies move from one scale to
the other, they tend to become segregated by traditional discipline
boundaries: cancer cell biologists do not necessarily communicate
with population geneticists, who communicate little with molecular geneticists, etc. [9]. For instance, it has been argued that, as
powerful as they are, molecular genetics studies often fall short of
fully understanding cancer genetics because they tend to disregard
how allele frequencies become fixed in a growing population of
heterogeneous cancer cells. That is, cancer molecular genetics has
traditionally neglected somatic cell population genetics [10]. For
this reason, the question as to whether mutated genes in cancer
cells are cause or effect of cancer progression remains a legitimate
one and to an extent unanswered. See, for instance, the ongoing
debate on mutation rates, i.e., whether or not a rate increase is necessary for progression [11]. From these distilled highlights that do
not of course do justice to the breadth of current cancer research, it
should be evident that considering the multiscale nature of cancer
progression is desirable. It is also worth mentioning why we think
it important to consider cancer progression as a complex process:
at every scale there are multiple variables that interact simultaneously. Therefore in order to make any useful predictions concerning
the mechanistic nature of cancer progression we propose to utilize
theoretical frameworks capable of integrating multiple variables at
once [12,13].
In summary, if cancer progression is a complex multiscale process, one effective means of comprehending it is to adopt multiscale
mathematical models that naturally incorporate multiple variables
across multiple scales. Accordingly, this approach has been adopted
by other disciplines (including physics, engineering, ecology and
economics) for forming testable theories of complex multiscale
processes. To be clear, there is no single mathematical model that
can encompass all aspects of cancer progression, due to the very
nature of its complexity. We therefore have been focusing on one
aspect of progression: the transitioning of tumor growth from
contained to locally invasive. Specifically, we consider three different multiscale mathematical models of invasion that bridge three
different spatial scales (but also overlap) from genes to cellular
phenotype to tissue.
3. Approaches and techniques to modeling cancer invasion
The occurrence of cancer invasion dramatically restricts
treatment options and is almost invariably fatal. Therefore, understanding and controlling invasion could have a major impact on this
dreadful disease. Invasion occurs at the tissue scale, but it is perhaps
best viewed as a property emerging from the collective behavior of
cells, at the scale below, in the context of a tumor mE. In fact, all of
cancer progression is best framed, for theoretical and experimental
purposes, in such multiscale fashion. The behavior of a single cell
is an emergent property of the relative balance of lower scale cellular core processes, including, e.g., division, growth, motility, and
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death. Each core process has properties emerging from the collective behavior of molecular signaling networks at the scale below.
The output of each signaling network derives from the behavior
of underlying genes. In this multiscale integrated framework, it is
easy to see how a mutation in a gene may propagate its effects up
the scales, ultimately determining the outcome of cancer progression, e.g., whether or not invasion occurs. The effects of a genetic
mutation on disease outcome (e.g., invasion) can also be modified by other variables at each scale. From this perspective, it is
inescapable that, to predict and curb invasion, we must have some
understanding of how multiple variables affect one another and
bridge the different scales. Complex multiscale processes such as
cancer progression (as discussed above) are effectively studied by
using mathematical modeling.
In more pragmatic terms, we have taken the view that, in order
to understand quantitatively and mechanistically invasion in the
context of cancer progression, top-down theoretical approaches
are needed to complement and organize the massive amount of
data and information produced by the wildly successful bottomup, molecular reductionist approaches [6]. We have proposed that
mathematical models should reflect the multiscale nature of cancer progression, and that particular care should be taken when
interfacing experiments with models, so that data and modeling
scales are properly matched [6]. Several models at distinct scales
are therefore necessary. However, in the specific case of invasion,
models operating at the cellular and tissue scale are particularly
relevant, because that is the level at which invasion is observed.
Here, we show our experience in building three different models,
EHCA, IBCell and HDC, which span scales from the genotype to the
tissue and in which the cell takes center stage. These models overlap, a highly desired property to easily interface the knowledge they
produce.
4. The mE in three mathematical models: EHCA, IBCell and
HDC
In this review, we focus on elements of these three mathematical models that concern the tumor microenvironment (mE). It is
perhaps useful to entertain both biological and mathematical considerations we have adopted in building these models. Regarding
our biological perspective, we are aware that it is customary to
consider the mE at the most intuitive level (stroma, ECM, vessels,
soluble factors, etc.), with little concern about scale separation. A
pervasive view is also that the many components of the mE should
be represented altogether, in order to capture their effects fully.
These views have been immensely helpful to unearth enormous
amounts of information. However, in our modeling efforts we found
it useful to sacrifice detail in order to distill essential elements of
the mE. For now, we are focusing on the following: space, mechanical forces, nutrients, proteases, and ECM. Each of these elements is
represented in a simple form that still provides sufficient realism
for simulations, but not simpler. For instance a generic form of ECM
or nutrients was adopted (see below for further details).
Our modeling approach is strictly reductionist (in the original
sense), and we strive to avoid a panglossian view of the mE [14]
in which every mE feature would have an evolutionarily optimized
purpose. To be clear, the models are built in such a way that the
represented elements of the mE interact with mathematical representations of cells or genes. In the case of cells, their perception of
the mE is an essential part of their phenotype. Simulation outcomes
are entirely determined by the interactions of cell phenotypes with
mE elements. In a sense, the outcome of interactions between cells
and mE in the models is entirely determined by rules that reflect
physicochemical laws. The simulations can be endowed with external assumptions based on biological knowledge. To the best of our
abilities, these assumptions do not predetermine outcome and are
suitable for experimental parameterization and refinement if necessary. Regarding our mathematical perspective, we appreciate that
it may certainly be possible to represent all of the many components of the mE at once in a model, in spite of the complexity of the
task. However, from a modeling perspective there is no particular
advantage in doing so—in fact it is best to adopt a minimal modeling approach because too many interacting variables and processes
reduce the likelihood of understanding the dynamics of the mechanisms that drive outcomes. Therefore, each of the three models
we describe includes a limited representation of mE components,
which in a sense matches the kind of process we are trying to
capture with that particular model, and its scale.
In the evolutionary hybrid cellular automata (EHCA) model, the
only mE variable is concentration of oxygen (as a generic representative of nutrients); in the immersed boundary cell (IBCell) model,
it is mechanical forces (both between cells and betweens cells and
the ECM) as well as oxygen concentration; in the hybrid discretecontinuum (HDC) model, the mE is reduced to nutrients (once
again, oxygen as a representative of), proteases and ECM. It is also
worth mentioning that in each model cancer cells themselves are
technically part of the mE since they affect the behavior of other
cells.
5. Brief description of the EHCA
The EHCA model uses a cellular automata based approach that
considers cells as simple grid points (Fig. 1(a)). Each cell contains
a complex neural network that links genotype to phenotype. The
grid itself represents the mE (see Fig. 1(a)) and the only variable on
this grid, apart from the cells, is the concentration of oxygen. The
dynamics of oxygen in space and time are controlled by a (continuous) partial differential equation. Cells on the grid consume oxygen
at a constant rate, whose value was derived from published experimental data [15]. The oxygen equation contains a term that links
cellular consumption with mE concentration. In the EHCA model,
cells perform three basic functions: proliferation, quiescence, and
apoptosis. The neural network within the cells controls these functions, via a feed-forward decision mechanism. The mE variables
(oxygen concentration and cell crowding, i.e., available space on the
grid) are considered as input to the genotype layer. Please note that
genotypes can vary and evolve from cell to cell, because of random
error at the time of copying (see below). The input is therefore processed by the cells’ individual genotypes in slightly different ways.
For instance, a value of oxygen concentration may trigger apoptosis
in one cell, but not another, because their respective genotypes have
evolved apart. Similar effects occur for quiescence and proliferation.
In aggregate, the genotype layer processes input, at the single cell
level, and then generates an output, also at the single cell level. This
output effectively represents phenotype, since it characterizes cell
behavior (proliferation, death, etc.) (see [16] for full details). In the
EHCA model, the mE is limited to the number of neighbors of a cell
and the local oxygen concentration. The level of cell crowding triggers competing behavior for space and will determine whether a
cell proliferates or becomes quiescent. More “aggressive” cells will
have evolved to become less prone to quiescence upon crowding.
The oxygen concentration decides the death response: again, more
aggressive cells have evolved to be less sensitive to death triggers.
At every cell doubling, the inner neural network is copied to the
daughter cells, but with a probability of error. This noise reflects,
in a sense, the occurrence of mutations. Of course, these mutations
modify how the genotype layer processes mE input, and distinct
outputs are generated, directly linked to the mutated genotype.
Therefore, the model captures one component of cancer progression, the generation of phenotypic variation within a population of
V. Quaranta et al. / Seminars in Cancer Biology 18 (2008) 338–348
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Fig. 1. Schematic of different model layouts: (left panel) a cluster of tumor cells in the EHCA and HDC models (colouration denoting cell state). The cells reside on a square
lattice and each lattice site can either be occupied by a cancer cell or be empty. Note, the EHCA model does not consider MDE (matrix-degrading enzymes) or ECM (extracellular
matrix). (Right panel) A portion of a small cluster of adherent cells in the IBCell model. Cell boundary points (dots) are connected by short linear springs to form plasma
membranes (grey lines); nuclei (circles inside cells) are surrounded by cytoplasm modeled as a viscous incompressible Newtonian fluid; cells are connected by adherent
links (thin black lines); adhesive and growth membrane receptors are distributed along the cell boundary.
growing cancer cells. It is of crucial importance to realize that the
effects of a mutation will directly alter the ability of a cell, and its
progeny, to compete for spatial a nutritional resources [16,17]. The
selective forces of the model can apply pressure on the phenotypes,
if they become limiting. For instance, decreasing levels of oxygen
concentration or increased cell crowding will produce a competition from which the fittest phenotypes will emerge. The genotype
of the fittest cells then increases its frequency in the total tumor cell
population. Therefore, in the EHCA model tumor growth is strongly
favored by adaptation to the mE, and the genotype/phenotype of
the resulting tumor is influenced by the selective strength of the
mE, which may or may not ignite competition among phenotypes.
For a detailed description of the EHCA model, refer to our recent
publication [16].
6. Brief description of IBCell
The tumor microenvironment exerts physical forces on cancer
cells [7,18,19] that affect their behavior. These forces are often overlooked in mathematical models of cancer progression. For our goal
of modeling cancer invasion at cell and tissue scale, however, it is
essential to represent this mechanical aspect of the tumor mE, since
it could capture critical steps in the initial stages of invasion, such
as breach of basement membranes, or distortion of epithelial structures (acini and ducts) that leads to local invasion and fingering.
To model mE mechanical forces, we used the Immersed Boundary method [20] to represent two-dimensional fully deformable
cancer cells. In our model (IBCell), the structure of each individual
cell includes an elastic plasma membrane modeled as a network
of linear springs that defines cell shape and encloses the viscous
incompressible fluid representing the cytoplasm and providing cell
mass. These individual cells can interact with other cells and with
the environment via a set of discrete membrane receptors located
on the cell boundary. In particular, each point can be engaged in
adhesion either with one of the neighboring cells or with the extracellular matrix (ECM). The ECM in this model is represented as
a viscous incompressible fluid. Fig. 1(b) shows a small cluster of
adherent cells with adhesive and free receptors distributed along
cell boundaries. Separate cells are connected by adhesive links that
are modelled as springs acting between boundary points of two
distinct cells if they are sufficiently close to one another. Several
cellular processes, such as cell growth, division, apoptotic death or
epithelial polarization, are modeled in IBCell. All individual cells
follow identically defined cell life processes. However, since cells
interact with one another, the initiation and progression of some
cellular processes may be a result of the cells’ collaborative or
competitive behavior. The mathematical formulation of cell life
processes that are crucial for the applications presented in this
section are described elsewhere [21]. In the current version of the
model, the focus is particularly on mechanical forces that cells exert
on each other. For the purposes of this review, therefore, one mE
influence we consider is the ability of cells to recognize the presence of other cells. For instance, if they are part of a lumen, they
may or may not sense apoptotic stimuli from outer cells. If they are
part of the outer layer of cells in acini, they may or may not polarize correctly. Another mE influence we consider is the background
oxygen level, like in the EHCA model. Depending on their hypoxic
tolerance, cells may undergo positive or negative selection.
If the cells are given an experimentally derived set of parameters,
they can go on to build realistic epithelial structures, such as acini,
ducts, and tubes [21]. It is important to realize that the outcome of
these simulations is not predetermined, but is an emergent property of the collective behavior of cells, i.e., the model is truly reductionist. The only external input to the model is the set of parameters
that regulate growth, proliferation, adhesion, death, and polarization. How cells use these parameters is purely determined by their
mechanical interactions with one another, and the mE. Therefore,
IBCell is ideal to capture the transition to invasion as epithelial
structures (acini and ducts) built or perturbed by cancerous cells
become unstable. For instance, we have used IBCell to simulate the
growing of cancer cells within the lumen of a duct, and the requirements for duct filling [22] and eventual breaching [22b]. Selective
pressure from the mE, in the form of low nutrient levels, will also
perturb the morphology of growing cell aggregates by selecting
only those cells that have maximum capacity to proliferate.
7. Brief description of HDC
The Hybrid Discrete Continuum (HDC) model operates at the
cell-to-tissue scale, and is broadly based on the growth of a generic
three-dimensional solid tumor, but, for economy of computation
time, we only consider a one-cell diameter thick two-dimensional
slice through it. Mathematically, however, translation to 3D is
straightforward [6]. Like the EHCA model, cells are considered as
points on a lattice and modeled using a hybrid cellular automata
approach. The mE consists of a two-dimensional lattice of extracellular matrix (ECM) (f) upon which oxygen (c) diffuses and is
produced/consumed, and matrix degrading proteases (m) are produced/used. The mE variables f, c and m are controlled by a system
of continuous reaction-diffusion equations whereas the tumor cells
(Ni,j ) are considered as discrete individuals which occupy single lattice points (i, j), hence HDC is a Hybrid between a Discrete and a
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Continuum model. Unlike the EHCA and IBCell models, cell movement is controlled by directional movement probabilities that are
functions of the local ECM concentration and allow the cell to
remain stationary (P0 ) or move west (P1 ), east (P2 ), south (P3 ) or
north (P4 ) one grid point at each time step. The motion of an individual cell is therefore governed by its interactions with the ECM
in its local environment. Of course the motion will also be modified by interactions with other tumor cells. Each individual cell
is also endowed with a predefined phenotype consisting of specific cell traits, including proliferation, death, cell–cell adhesion,
cell–matrix adhesion, and secretion of ECM-degrading proteases
(m), which determines how it behaves and interacts with other cells
and its environment. These traits can be parameterized with experimental data from the laboratory or the literature. A more detailed
description of the model and its application have appeared in print
[17,23]
A key feature of the HDC model is that the tumor cell population is heterogeneous with each cell phenotype being defined from
a pool of 100 randomly predefined phenotypes within a biologically relevant range of cell-specific traits (more or less than 100
phenotypes could easily be considered, likely with similar outcomes). In order to capture the process of mutation we assign cells
a small probability (Pmutat ) of changing some of their traits at the
time of cell division. If such a change occurs, the cell is randomly
assigned a new phenotype from the pool of 100. A feature of the
HDC model especially relevant to invasion is that cells have the
ability to move. This makes it possible to examine (and parameterize from experiments) the impact of cell–ECM interactions on
cell migration. In summary, in the HDC model cell migration, proliferation and death, coupled with mutating phenotypes, create a
tumor population whose individual cell components interact with
each other and have the ability to adapt to the mE (within the confines of the 100 randomly defined phenotypes). In this scenario, it
is possible to examine tumor growth morphological outcomes, the
traits of phenotypes best adapted to a local mE, the dynamics of the
adaptation process, and the influence of a range of different mE on
morphology and adaptation outcomes.
8. Distinctive mE modeling features of EHCA, IBCell and
HDC
By limiting the mE in all three models to only a few elements
we have been able to better focus on their effects on cells. Thus all
three models are aligned with our ultimate goal, which is to identify
fundamental mechanisms that underlie interactions between cells
and their mE during cancer invasion. Nonetheless, in each model
the mE has some differences worth noting.
In EHCA, the mE elements are represented as information input
to a network that links genotype with phenotype. In a sense, this
information (together with random mutations) can be viewed as
input that trains the network to adapt. This is an abstraction of a
fundamental biological process, cell adaptation to mE conditions.
Perhaps the closest experimental method that captures this process
is gene expression analyzed by microarrays [24]. In this technique,
cells are placed in distinct mE (e.g., high and low serum concentrations), and their gene expression profile is compared, globally [25].
A key difference between EHCA and microarrays is that in EHCA
the gene layer network is monitored at the single-cell level, whilst
in microarrays, for technical reasons of sensitivity, gene expression
profiles are the average of a population of cells (although ultrasensitive single-cell microarrays will be available, no doubt, in the
future). We are exploiting these similarities to improve analysis
of microarray data with new analytical tools (Gerlee and Zhang,
in preparation). This may become an excellent example of theory
guiding experimentation and vice versa, since the single-cell set up
of EHCA can deconvolve the gene expression averages of microarrays, whilst microarray expression data can validate theoretical
predictions of the EHCA.
In IBCell, our intention was to represent the mE primarily as
mechanical forces. In a sense, we wanted to represent the interface between chemical and physical events, based on the view that
the physical laws of the mE (which includes cells) determine and
constrain cell behavior, which is based mostly on chemical reactions (of course, there are chemical reactions occurring in the mE
as well). There is a high level of abstraction in this model, e.g., receptors are grouped in broad categories, unlike real life, and they all
directly transduce mechanochemical information. IBCell is aimed
at representing the effects of the mE on intercellular organization
and the resulting creation of morphological structures that make
functional sense. In cancer, the morphology of histological structures made by transformed cells has been described as a caricature
of normal histology [26,27]. IBCell has the potential to quantify the
influence of the mE on the process of morphogenesis, particularly
epithelial morphogenesis [22] in both normal and cancerous tissue.
In HDC, the mE contains more elements, inclusive of soluble
nutrients, ECM and matrix degrading enzymes. HDC has the least
level of abstraction, and in fact it has the potential to explicitly
incorporate mE elements from real life, such as inflammatory cells,
angiogenesis, stem cell niches, etc. The goal of HDC, in a sense,
is to determine how single-cell behavior translates into emergent
properties at the cell population level. In biological terms, one
can think of it as cell differentiation affecting tissue organization,
because in HDC the cellular phenotype is a composite of traits that
explicitly reflect core cell processes, such as proliferation, migration, metabolism, secretion, etc. By dialing each of these traits up
or down, in different combinations, HDC effectively defines differentiation states of every cell in the simulation. For instance,
HDC could model epithelial to mesenchymal transition, or vice
versa.
From this brief description of the distinct features of the three
models, we hope it is clear that they operate at different but overlapping scales, though all considering the cell as the fundamental
unit: EHCA is at the molecular (gene expression) scale, IBCell is
solidly at the cell scale, whilst HDC captures primarily the tissue
scale.
Common features in the three models make it possible to compare their inner workings, outcomes, predictions or conclusions.
They can be highlighted as follows: (i) each model treats cells
as individuals with their own distinct inner life processes (e.g.,
proliferation, death, mutation, and migration). (ii) Cells sense and
influence each other (e.g., death signals in IBCell), and interact with
and alter the local mE, e.g., by occupying space, sensing nutrients
and modifying their concentrations by consumption. (iii) Acquisition of two resources, space and nutrients, is the major influence
on adaptive behavior of cells. (iv) In all three models, since they are
individual-cell based, cells themselves are part of the mE, e.g., the
number of neighboring cells is a variable that contributes to modifying proliferative response, akin to contact inhibition. (v) Availability
of resources in the mE sets the stage for competition between individual cells during adaptation and can set in motion a struggle
among phenotypes that evolves the tumor population.
We now examine simulations from these models in a range of
mE conditions, and discuss the implications of outcomes.
9. Insights from simulated tumor growth dynamics
9.1. Fingering morphology as a modeling abstraction of invasion
In order to understand the key role that nutrients play in driving
changes in tumor morphology we use each of the three models to
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Fig. 2. Simulation results from all three models for tumors grown under a range of mE background oxygen levels (k) and cell hypoxia tolerance thresholds (h): HDC (upper
figure in each square), EHCA (left figure in each square) and IBCell (right figure in each square). Both parameters (k and h) influence the morphology of the tumor: increasing
k reduces size and produces fingering, increasing h has a similar, albeit more subtle effect. Colouration for cell types: red (proliferating); green (quiescent); blue (dead).
Fig. 3. Growth simulation outcomes from the IBCell model. (Upper row): Spatial distribution of cells after 11 cell divisions. Under different mE and hypoxia tolerances,
three distinct growth patterns are observed: (left) solid tumors (nutrient-rich mE, normal hypoxia tolerance), (center) tumors with fingering margins (nutrient-scarce mE,
normal hypoxia threshold), (right) tumors in growth arrest (nutrient-poor mE, low hypoxia tolerance). (Lower row): distribution of all cells according to the concentration
of nutrients sensed by each cell and the percentage of free growth receptors. Horizontal lines represent the hypoxic tolerance level. Vertical lines represent the 20% growth
threshold. Cells colouration denotes cell state: growing (red), quiescent (green), hypoxic (blue).
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investigate how changes in cell metabolic parameters influence the
structure of the developing tumor. Specifically we consider a single
mE value, oxygen concentration. Changes in this mE variable are
sensed by cells and if the hypoxia threshold level h occurs, they
become growth arrested or die. For all three models we start each
simulation with the same initial cluster of tumor cells embedded in
a homogenous field of a given oxygen concentration and allow cells
to interact with their mE. This leads to the development of different
tumor morphologies including the emergence of tumor fingering,
summarized in the form of a graphical table (see Fig. 2).
Several trends are apparent (Fig. 2) in the resulting tumor
morphologies from representative simulations in which we systematically varied availability of mE oxygen and cell hypoxia
sensitivity (h). A common feature is that living (red) cells are mostly
located on outer rims and surround central cores of dead (blue)
cells, consistent with the idea that the best adapted phenotypes
are the ones capable of positioning themselves at the tumor margin, by proliferation or motility. For high mE oxygen levels and
low hypoxia thresholds (cells require less oxygen to survive or
proliferate), the tumor grows into a compact morphology, with
a smoothly growing leading margin that leaves an even distribution of dead cells in its wake. As the background oxygen level is
decreased and the hypoxic cell threshold is elevated, a fingering
morphology emerges. Decreasing mE oxygen levels also appear to
decrease the overall size of the tumor. In contrast, elevating hypoxia
thresholds result in approximately constant tumor size, but emerging fingers take on thinner, more elongated shapes. These trends
are best visualized along the diagonal (Fig. 2, top left to bottom
right): simulated tumors are smaller and more fingered as mE oxygen decreases and oxygen requirement by cells increases. These
results point to a direct correlation between tumor morphology and
the intensity of competition for resources during simulated tumor
growth.
Whilst all three models produce similar trends in tumor morphologies under similar conditions, there are also some differences.
The HDC and EHCA models produce the closest results to each
other. However, the EHCA simulated tumors are generally much
more compact, mostly due to the fact that the cells can migrate
in the HDC but not in the EHCA model (movement is currently
being added, Gerlee and Anderson (submitted)). The IBCell model
results present some unique features that can be grouped into
three distinct morphologies: large clusters with smooth boundaries, medium size clusters with finger-like extensions and small
clusters of non-growing cells. The reason for these differences from
the other two models is due in part to scale and in part to the
limited ability of cells in IBCell to evolve by progressive adaptation. Nonetheless, if we examine the three outcomes specific to
IBCell simulations in more detail, a possible reason emerges for
the differences in model outcomes as well as tumor morphologies. Fig. 3 (upper panel) shows the distribution of cells for the
three different IBCell simulation outcomes: solid, fingered, growth
arrested. As the IBCell model is driven by receptor dynamics, we
examine the distribution of cells in each of these morphologies
according to their percentage of free growth receptors and the total
concentration of oxygen sensed by each cell. Fig. 3(a), shows that
the majority of cells above the growth threshold of free growth
receptors are actually growing (red), all quiescent cells (green) lie
in the region above the hypoxic threshold and below the growth
threshold, and almost all hypoxic cells (blue) are starving and are
overcrowded. In contrast, Fig. 3(b) shows that numerous hypoxic
cells lie above the growth threshold, that is they maintain more
than 20% of their receptors free from cell–cell adhesion, so their
growth suppression is due to low levels of oxygen in their vicinity
and not to overcrowding, this suggests that the finger-like morphology is rather a result of hypoxia-related growth suppression than
overcrowding. Fig. 3(c) shows in turn, that all cells in the growth
arrested tumors lie well below their hypoxia threshold. In spite of
this uniqueness, the common trend persists in IBCell: emergence of
a fingering morphology in the presence of low oxygen conditions,
i.e., a harsh mE that can spawn competition among adapting cell
phenotypes.
9.2. mE influence on tumor evolutionary dynamics
The mE not only affects tumor morphologies in the simulations
discussed above but also acts as a selective pressure that leads to the
clonal dominance of more aggressive phenotypes. To understand
Fig. 4. Growth simulation outcomes from the EHCA model. (Left panels): Spatial distribution of cells in an oxygen switching experiment. In high oxygen mE, the tumor
consists mostly of quiescent cells and grows with a round morphology, whilst in low oxygen mE the tumor is dominated by dead cells and displays a fingering morphology
(outcomes of two independent simulations are shown). (Line graph on the right): Time evolution of the phenotypes in the simulations on the left panels. The most abundant
phenotypes have been highlighted and their response vectors are displayed—the vector takes the form of three probabilities for (proliferation, quiescence, and apoptosis), so
that (1,0,0) means the probability of proliferation is 1, i.e., it will always occur.
V. Quaranta et al. / Seminars in Cancer Biology 18 (2008) 338–348
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Fig. 5. Growth simulation outcomes from the HDC model under three different mE: (A) uniform ECM, (B) Grainy ECM and (C) Low nutrient. (Upper row): spatial tumor
cell distributions after 3 months of simulated growth shows that the three different microenvironments have produced distinct tumor morphologies. In particular, the
homogeneous ECM distribution has produced a large tumor with smooth margins (A) containing a dead cell inner core and a thin rim of proliferating cells. The tumor in the
grainy ECM also has a dead inner core with a thin rim of proliferating cells, however, it displays a striking, branched fingered morphology at the margins (B). This fingering
morphology is also observed in the low nutrient simulation, which produced the smallest tumor (C). (Lower row): relative abundance of the 100 tumor phenotypes as the
tumor grew in the different mE: approximately six dominant phenotypes in the uniform tumor, two in the grainy and three in the low nutrient tumor. These phenotypes
have several traits in common: low cell–cell adhesion, short proliferation age, and high migration coefficients. In each tumor, one of the phenotypes is the most aggressive
and also the most abundant, particularly in B and C. All parameters used in the simulations are identical with the exception of the different mE.
this result we will examine each of the model results in more detail
and under different mE conditions.
The prediction that a harsh environment induces tumor fingering raises the question: is it possible to reverse the fingering
morphology by simply increasing the available oxygen levels? If
so, how will this affect the growth and evolutionary dynamics of
tumor populations? To investigate this point, we focus on the EHCA
model and impose two different forms of competition upon the
tumor cells—for space and nutrients in the poor oxygen mE (harsh),
and for space only in the oxygen rich mE (mild). Fig. 4(a) shows
simulation results in which the mE is switched between the two
conditions. Consistent with results obtained in Fig. 2 the simulated tumor grows with smooth margins in a mild mE, whereas
when switched into a harsh mE, finger-like protrusions immediately occur. Analysis of the phenotype dynamics (Fig. 4(b)) reveals
that the phenotype composition of the population also changes
under the different mE conditions. In particular, we observe the
emergence of more aggressive phenotypes in the oxygen starved
mE (e.g., the {1, 0, 0} and {0.9, 0.1, 0} phenotypes).
The implication from this result is that tumor fingering goes
hand in hand with the selection of more aggressive phenotypes.
However, if we were to consider different harsh mE conditions
would the same outcome result? As we reported previously, using
the HDC model [23] we can investigate different mE conditions and
answer this very question. Fig. 5 shows the results of growing in silico tumors under three different mE conditions, (A) uniform ECM
and normal nutrient, (B) grainy ECM and normal nutrient and (C)
uniform ECM and low nutrient. Both of the mEs in (B) and (C) could
be considered to be harsh as they are more difficult mE to grow
in (Fig. 5, upper row). Similar to the results from the EHCA model,
tumor growth in these harsh conditions leads to both a fingered
morphology and fewer, more aggressive cellular phenotypes in (B)
and (C) (Fig. 5, lower row). The aggressive phenotypes obtained in
both the HDC and EHCA models under harsh mE growth conditions
are similar in that they have the highest proliferation rates; however, HDC phenotypes are more detailed and reveal that the most
aggressive phenotypes also have low cell–cell adhesion and high
cell motility values.
The EHCA and HDC models produce consistent results; however
both models begin with cells that have already undergone some
initial step of tumorigenesis, e.g., growth is unrestrained. By contrast, the IBCell model is initiated with normal epithelial cells and
can progress without phenotype mutations to produce epithelial
structure. However, IBCell can be used to test diverse scenarios for
the very early steps of tumor initiation, e.g., growth inhibition by
mE forces is reduced in all cells, or an occasional cell within the
epithelial cluster does not respond to mE growth inhibition cues.
Fig. 6 shows the IBCell simulated development from a single epithelial cell to: (i) a normal acinus, or (ii and iii) two aberrant structures.
In Fig. 6ii, luminal filling is achieved by removing response to mE
growth inhibition cues from all cells (interestingly an acinus of
similar size and shape to normal structure forms, though with
a filled lumen). In Fig. 6iii, distorted acinus morphologies, reminiscent of invasive phenotypes, occur via mutation of a single
cell such that it loses its ability to polarize and become growtharrested.
In summary, three models, all individual cell based but very different in their construction, can be used to analyze the effect of mE
variables on tumor growth. Strikingly, they all reach the same conclusion, and point to competitive adaptation to mE conditions as a
determining factor for invasion: both invasive tumor morphology
(“fingering”) and evolution of dominant aggressive clonal phenotypes appear to occur by a process of progressive cell adaptation to
mE’s that support sustained competition between distinct cancer
cell phenotypes. Currently, we are engaged in testing these models’
general predictions via both parameterization of the HDC (Ander-
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Fig. 6. IBCell simulation outcomes. (i) Consecutive stages in the development of a normal hollow acinus from a small cluster of viable cells to an acinar structure composed
of a complete outer layer of polarized cells surrounding a hollow luminal space; (ii) formation of an abnormal acinus with a complete outer layer of polarized cells and
an inner core of cells that fail to become growth suppressed and fill the luminal space; (iii) formation of an abnormal acinus with a cohort of invasive cells arising from a
polarization-deficient precursor cell, which deform the outer layer and break through to the outside and to the luminal space. Numbers indicate completed rounds of cell
divisions. Cell phenotypes are coloured as follows: resting viable cells (yellow); growing cells (green); polarized cells (red); apoptotic cells (grey); dead cells (black dots).
son et al., submitted,) or IBCell (Rejniak et al., submitted) models,
and validation (Kam et al., in press; Rejniak et al., submitted).
9.3. Adaptation and selection during cancer progression
In this review, we have presented three mathematical models of cancer invasion, each representing the mE in different, but
connected ways:
(a) in EHCA, the mE is represented as molecular concentrations of
nutrients or factors: this limited abstraction appears to be the
most appropriate for the purposes of EHCA, to model the link
between genes and phenotypes in distinct mE.
(b) in IBCell, the mE is essentially mechanical forces, because the
goal of IBCell is to model mechanotransduction, i.e., how cells,
as deformable objects driven by chemical reactions, aggregate
(at the scale of a few hundred) and form meaningful functional
multicellular structures interfacing the physicochemical mE.
(c) in HDC, mE influences include both neighboring cells and concentrations of oxygen or ECM macromolecules, in order to
model large scale tissue formation (millions to billions of cells)
from single cell behavior.
The predictions of all three mathematical models converge on
one message: invasion emerges from the population dynamics of
single cancer cells growing and adapting to the mE, when a harsh
mE ignites competition for resources between phenotypes with
distinct adaptive features. Whilst the models do not disprove the
possibility that the mE may also have a regulatory (instructive)
effect on cancer cells [18], or that some cancer cell phenotypes
may be intrinsically invasive [28], nonetheless they place a major
emphasis on another proximate cause for invasion: the struggle
among cancer cell phenotypes with distinct adaptive values to
survive by co-opting mE resource(s) that have become scarce. It
is important to realize that, in the models, distinct phenotypes
are generated at random, whilst their relative frequency is nonrandom and based on their adaptive value to the mE [16,23] (fitness,
intended as the ability to increase their frequency in the next time
step). Therefore, in the models two elements appear to be essential
for invasion: adaptation and competition. If either is missing, neither morphological invasion (fingering) nor phenotype evolution
may proceed in the models. To be clear, if adaptation is abolished
(e.g., by dialing “mutation rates” to zero in the HDC model), cancer
cells still form tumors if the mE is favorable, but they consistently
present smooth, non-invasive margins and, naturally, evolution
by competition is absent because the cancer cell population is
immutable (with the exception of an extremely aggressive phenotype matched to an extreme mE, in which case the aggressive
phenotype, though unchanging, competes with itself and fingers).
If, on the other hand, competition is abolished from the simulations,
aggressive phenotypes do emerge from the adaptation process,
but in the non-competitive mE they neither invade nor dominate
and coexist with other adapted phenotypes in a smooth-margined
tumor. In summary, the mE enables invasion by creating competitive conditions among adapting phenotypes [29].
9.4. Parallels between cancer progression and Darwinian
evolution
Cancer progression to invasion is described by our models as
a process of competitive adaptation ignited by a harsh mE. This
framework has obvious similarities to a classic (Darwinian) process of evolution by natural selection. Parallels between cancer
progression and Darwinian processes have been discussed recently
[27,30–33]. However, caution should be exercised in jumping to
conclusions. This caution is justified by insufficient quantitative
data from real or experimental tumors regarding the three key
components of Darwinian evolution: variation, inheritance, and
selection. Limitations to be overcome include the following.
Variation: available methods are powerful but there are at least
two problems [34]: (1) most of them average out information on
genotypes (gene expression profiling) or phenotypic traits, whereas
information on single-cell phenotypic plasticity, variability of phenotypic traits within isogenic cells, and quantitation of phenotype
among clonal cells with distinct genotypes is desirable, if not absolutely necessary; (2) the mapping of genotype to phenotype is
V. Quaranta et al. / Seminars in Cancer Biology 18 (2008) 338–348
347
Fig. 7. A schematic plot of cancer evolution as a function of time, driven by two distinct adaptive strategies. Initially a well-adapted phenotype is pushed to operate at one
extreme of its plasticity range, in order to adapt to changes in the mE. A random genetic mutation in its progeny stabilizes this effect and creates a phenotype well adapted
to the new mE. This process then repeats and leads to further rounds of phenotype adaptation and genotype stabilization (possibly based on additional mutation events),
producing a mix of phenotypes with distinct adaptation and adaptability values. At any point in this process, invasion may emerge if a harsh mE (e.g., nutrient- or oxygen-poor)
develops and drives competition between these phenotypes (see Fig. 2).
obviously not one-to-one (one gene and one trait), and therefore
experimentally translating genotypes into phenotypes (a collection of behavioral traits, such as proliferation rate, migration rate,
metabolic rate, etc.) is a must, because mE selective forces act on
these traits.
Inheritance: the need for theoretical, quantitative treatment of
somatic cell population genetics has been repeatedly advocated
[9,10]. We subscribe to that view: without this framework, it will
be impossible, for one, to connect properly observed frequencies of
mutations in cancer cells with disease outcome.
Selection: whilst the importance of the mE in determining cancer
progression is now recognized, quantitative measurements of mE
parameters (concentration of nutrients or mitogens, oxygen tension, physical forces, etc.) are still in their early days and far from
being routinely applied. In fact, technology for such measurements
is in need of development.
Once quantitative information on these three aspects of cancer
reaches a critical mass, it will be more feasible to determine to what
extent Darwinian selection applies to cancer progression. There is
hardly any doubt that some form of adaptation, or even evolution,
of cancer cell phenotypes occurs during cancer progression: The
concept of “clonal selection” [35] is a cornerstone of cancer biology.
However, it is perhaps worth reminding ourselves that evolution by
natural selection is by no means the only mechanism of adaptation
to the mE, as elegantly summarized by Gould and Lewontin [14]:
“The mere existence of a good fit between organism and environment is insufficient for inferring the action of natural selection”.
For instance (Fig. 7), on the longer time scale (years or decades)
necessary for cancer to arise, it is possible that classic Darwinian
processes, operating in forming cancer tissue, may overcome the
rate-limiting steps proposed by multistage progression theory [36]
(developed to explain age-specific cancer incidence data [10,36]).
In contrast, for the shorter time scales (weeks or months) of
terminal cancer progression (Fig. 7), other adaptation/evolution
mechanisms may be operational. Epigenetics [37,38] come to mind.
Another possibility (Fig. 7) is adaptation to stressful mE’s that
favors genetic changes [39]. Mathematical models such as the ones
described here can, if properly parameterized, provide powerful
theoretical frameworks for experimental validation of their predictions about mechanisms underlying cancer progression at various
stages. Of course, experimental parameterization and validation is a
task far from simple. As we discuss elsewhere [6], it requires close
integration between theoreticians and experimentalists. Further-
more, the enormous amount of data and technology in modern
experimental biology is a critical wealth, but it will need to be
complemented by mathematics-driven experimentation and novel
validation-enabling techniques [6].
10. Conclusions
We hope to have justified how a theoretical framework rooted in
mathematical modeling can provide a multiscale, unifying view of
cancer progression, with both explanatory and experiment-driving
power. Understanding mechanistic details of individual variables
of cancer progression is highly desirable and, rightly, most of the
cancer research enterprise is focused on this task. This molecular
reductionist approach should continue unabated. However, its very
success, its ability to produce enormous amounts of data at ever
accelerating rates, especially at the molecular scale, has created a
huge opportunity and an urgent need for integrative theoretical
approaches. Mathematical modeling coupled with experimentation is one powerful approach to fulfill this opportunity and need.
For example, the computational results presented here on the
importance of adaptation (operating in different modes) would be
difficult to grasp intuitively. Mathematical models provide a theoretical framework that produces quantitative testable hypotheses.
Experimentation then becomes essential in order to verify (1)
whether simulation outcomes apply to real tumors; (2) which
mechanisms drive the various stages of tumor progression.
Finally, there is one intriguing conclusion that is worth highlighting because it complements current translational research. It
is becoming a general view that cancer is not one, but many diseases
that must be characterized individually. In fact, in this genomic era
of personalized medicine, it is often argued that even a given type of
cancer, e.g., breast or lung cancer, is in fact a composite of many diseases that must be deconvolved at the level of individual patients,
for successfully applying treatment [40,41]. Our models offer an
alternative perspective: cancer is one disease, caused by a runaway
process of evolution by natural selection, which gives progression
its ruthless direction towards malignancy. The outcome of this one
disease, however, is diverse in each patient, because its driving process contains stochastic components. That is, one disease, but not
the same in every patient. Personalized cancer treatment is still
the road to follow [42], guided however by the quantitative perspective of properly parameterized computational models that can
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V. Quaranta et al. / Seminars in Cancer Biology 18 (2008) 338–348
make sense of both the deterministic and stochastic aspects of the
disease, as it arises in a specific patient. It will be fascinating to
witness the reach and direction of cancer research as guided by
theoretical frameworks.
Acknowledgments
Work described in this review was supported by grant
U54-CA113007 from the Integrative Cancer Biology Program
of the National Cancer Institute. The authors are grateful
to Lourdes Estrada and Alissa Weaver for thoughtful reading of the manuscript, and to all members of the VICBC
(http://www.vanderbilt.edu/VICBC/) for their enthusiasm and hard
work.
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